Algebraically Complete Field

algebra

Definition

Algebraically Complete Field

A field $F$ is algebraically complete (or algebraically closed) if every non-constant polynomial with coefficients in $F$ has at least one root in $F$.

Formally: for every $p\in F[x]$ with $\mathrm{deg}(p)\ge 1$,

$$\exists \alpha \in F:p(\alpha )=0.$$

Equivalently, every $p\in F[x]$ splits into linear factors over $F$:

$$p(x)=c\prod _{i=1}^{\mathrm{deg}(p)}(x-{\alpha }_i),c,{\alpha }_i\in F.$$

Examples

$\mathbb{C}$ is algebraically complete

By the Fundamental Theorem of Algebra, every non-constant polynomial with complex coefficients has a complex root. Hence $\mathbb{C}$ is algebraically complete.

$\mathbb{R}$ is not algebraically complete

The polynomial $p(x)=x^2+1\in \mathbb{R}[x]$ has no real root, since $x^2+1>0$ for all $x\in \mathbb{R}$.